Random variable X is a mapping that maps each outcome s in the sample space to a unique real number x, < x <. ( ) X s. Real Line

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1 Random Variable Random variable is a mapping that maps each outcome s in the sample space to a unique real number, < <. ( ) s s : outcome Sample Space Real Line Eample Toss a coin. Define random variable as follows: = 0 if heads if tails Roll a dice. Define random variable Y as follows: if,3,or 5 Y = 0 if 2,4, or 6 Consider a packet router. Eamine its queue at a random time. Define random variable N as the number of packets waiting in the queue. Go to the bus stop at a random time. Define random variable W as the amount of time you wait until the net bus arrives. Each time we repeat the eperiment, the outcome may change in the sample space and the value of the random variable changes according to the rule of the mapping, Notes on the Mapping Range of random variable : R = { = ( s) for some s S} In general, is a many-to-one mapping, but never one-to-many

2 2 Eample- Bernoulli Toss a biased coin. The coin falls down heads with probability p. Define random variable as = 0 for heads for tails Range: R =? P= [ ] =? P [ = 0.5] =?, P [ 0.] =?, P [ > 0] =?, P [ 2] =?, P [ > 3] =?. is referred to as a Bernoulli rv. Eample - Geometric Toss a biased coin. is the number of times we toss the coin until we see the first head. Range: R = {,2,3, } P [ = j] =? is a referred to as a geometric rv. Eample Eponential Consider a packet router that passes arriving packets to net destination routers. Measure the time between successive packets arrivals, referred to as the packet inter-arrival time. Let denote the packet inter-arrival time. Range: R = { r 0 r< } Eperiments show frequently [ ] λ P [ ] e for 0, where λ is the packet arrival rate, e.g., packets/sec. is referred to as an eponential rv.

3 3 Use of Random Variables With rvs, we can define functions of rvs. For eample, Find Pe [ ] U = + Y. Find P U. [ ] W = Rsin Θ. Find P W w. Notation P[ = ] : is a random variable, and is a constant or a simple variable. Types of Random Variables Discrete rv Continuous rv Discrete Random Variables The range consists of finite, countable real numbers such as{4,6,8}, or countably infinite real numbers such as {0,,2, } or {, 2,,0,,2, }. Continuous Random Variables When the range is not countable, the random variable is a continuous one.

4 4 Probability Distributions cumulative distribution function (cdf) probability density function probability mass function cumulative distribution function (cdf) The cdf of a random variable F ( ) = P [ ] is defined as Eample Uniform Distribution Throw a dart at a spinning wheel. is the phase where the dart hits the wheel. The range of the random variable is R = {0 < 2 π}. Within the range, 2 P [ 2] = for any phases π. 2π The cdf is 2π F ( ) = P[ ] = 0 < 2π 2π 0 < 0 F ( ) 0 2π is a continuous random variable. is referred to as a uniform random variable, or is said to have a uniform probability distribution.

5 5 Eample Uniform Distribution Buses arrive periodically with a period of T. You arrive at the bus stop at a random time. Define random variable W as the time you wait until the net bus arrives. Find the cdf of W. Ans. > T FW ( ) = P[ W ] = 0 T T 0 < 0 Uniform Distribution ( ) In short, U a, b, a< b. b a F ( ) = P[ ] = a < b b a 0 < a Eample - Bernoulli Toss a coin. Define as for a head with prob p = 0 for a tail with prob p. is referred to as a Bernoulli random variable. F ( ) = P[ ] = p 0 < 0 < 0 F ( ) p 0 All discrete rvs have discontinuities in their cdf. The value of the cdf is taken approaching from the right.

6 6 Properties of cdf 0 F ( ) 2 F ( ) = 3 F ( ) = 0 4 F ( ) is a non-decreasing function of 5 F ( ) is continuous from the right: that is, F ( b) = lim F ( b+ h). h 0 6 P[ a< b] = F ( b) F ( a) However, for a continuous rv, feeel safe to say Pa [ b] = F ( b) F ( a)

7 7 probability density function (pdf) The pdf of a rv is the derivative of its cdf: d f( ) F( ) d Properties of pdf The pdf is the rate at which the cdf increases. 2 Integrate the pdf to get the cdf. F( ) = f( u) du 3 To find the probability b P[ a b] = F ( b) F ( a) = f ( u) du a Be careful of the = sign for discrete random variables. 4 f ( ) 0 for any, and f ( u) du=.

8 8 Eample Y is an eponential random variable with thee cdf FY ( y) = e [ ] i) Find P Y 2. 2 y for y 0. ii) Find the pdf of Y. iii) Plot the cdf and pdf. For any pdf, i) non - negative ii) area sums to. For any cdf, i) nonn - decreasing from 0 to

9 9 probability mass function (pmf) The cdf of a discrete random variable has discontinuities. F ( ) p 0 The pdf consists of delta functions. f ( ) p p 0 When the range is {,,, }, we often use the notation p P[ = ]. 0 2 { p k } of a discrete random variable is referred to as the probability mass function (pmf). k k For a discrete random variable, it is easier to find the pmf first and then the cdf from the following relation F ( ) = P[ ] = p. k k

10 0 Notes on Continuous Distributions For a continuous random variable, it is easier to find the cdf first and then find the pdf by differentiating the cdf. d f ( ) = F( ) d 2 It is wrong to say P[ ] f ( ) = = for a continuous random variable. For a continuous random variable, P[ = ] = 0 for any. However f ( ) may not be zero. Then how are P[ = ] and f ( ) are related? Ans. Note for a continuous rv, For a small Δ, [ ] = ( ) Pa b f ( d ) for any inerval ab,. [ ] P +Δ f ( ) Δ. ( c) Eq. c is often used for finding the pdf directly. b a

11 Roots of Popular Probability Distributions

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